1. The first step is to identify this as a geometric series.
Remember you can calculate the sum of a geometric series using the formula:
[tex]S_n=\frac{a(r^n-1)}{r-1}[/tex]Where:
• S, is the sum
,• a, is the first term
,• r ,is the common ratio (what you're multiplying by to get to the next term)
,• n ,is the number of terms you're adding up
In this case, we have to calculate wich n corresponds to 163840, using the geometric sequence:
[tex]T_n=ar^{n-1}[/tex]This way,
[tex]\begin{gathered} 163840=(5)(2^{n-1})\rightarrow163840=(5)(2^n)(2^{-1}) \\ \rightarrow\frac{163840}{2^{-1}}=(5)(2^n)\rightarrow327680=(5)(2^n) \\ \rightarrow\frac{327680}{5}=(2^n)\rightarrow2^n=65536 \\ \rightarrow n=\log _2(65536)\rightarrow n=16 \end{gathered}[/tex]Now, let's use the formula for S:
[tex]S_n=\frac{5(2^{16}-1)}{2-1}\rightarrow S_n=327675[/tex]
